The properties of the entanglement in the intensity-dependent coupling interaction between squeezing vacuum state field and atom with an additional Kerr medium are investigated by using the von Neumann entropy. The influences of nonlinear interaction intensity of Kerr medium and the squeezing degree of initial squeezing vacuum state field on the entanglement of the quantum system are discussed. The results show that the nonlinear interaction of Kerr medium can result in the generation of the maximally entangled state, disentangled-free state or disentangled state, and the degree of entanglement is stationary.